Let $\displaystyle A$ be a group with normal subgroups $\displaystyle M$ and $\displaystyle N$.

Let $\displaystyle a \in A$.

Suppose $M \cap \langle a \rangle =1$ and $N \cap \langle a \rangle= \langle a^{s} \rangle$ for some positive integer $\displaystyle s$.

Can we find $MN \cap \langle a \rangle$???